1. Models and Algorithms for Complex Networks
Theory and Algorithms for Link Analysis Ranking, Rank Aggregation, and Voting

2. Outline
Axiomatic Characterizations of Link Analysis Ranking Algorithms
InDegree algorithm
PageRank algorithm
Rank Aggregation
Computing aggregate scores
Computing aggregate rankings - voting

3. Comparing LAR vectors How close are the LAR vectors w1, w2?

4. Distance between LAR vectors
Geometric distance: how close are the numerical weights of vectors w1, w2?
w1 = [ ]
w2 = [ ]
d1(w1,w2) = = 1.6

5. Distance between LAR vectors
Rank distance: how close are the ordinal rankings induced by the vectors w1, w2?
Kendal’s τ distance

6. Similarity Definition: Two algorithms A1, A2 are similar if
Definition: Two algorithms A1, A2 are rank similar if
Definition: Two algorithms A1, A2 are rank equivalent if

7. Monotonicity
Monotonicity: Algorithm A is strictly monotone if for any nodes x and y y
wx < wy x

8. Locality
Locality: An algorithm A is strictly rank local if, for every pair of graphs G=(P,E) and G’=(P,E’), and for every pair of nodes x and y, if BG(x)=BG’(x) and BG(y)=BG’(y) then
the relative order of the nodes remains the same
The InDegree algorithm is strictly rank local G G’

9. Label Independence
Label Independence: An algorithm is label independent if a permutation of the labels of the nodes yields the same permutation of the weights
the weights assigned by the algorithm do not depend on the labels of the nodes

10. Axiomatic characterization of the InDegree algorithm [BRRT05]
Theorem: Any algorithm that is strictly rank local, strictly monotone and label independent is rank equivalent to the InDegree algorithm

11. Proof outline Consider two nodes i and j with d(i) > d(j)
Assume that w(i) < w(j)
|R| = |L| L C R E
|E| > 0
graph G j i

12. Proof outline Remove all links except to i and j
w1(i) < w1(j) (from locality) L C R E j i
graph G1

13. Proof outline Add links from C and R to node k
w2(i) < w2(j) (from locality)
w2(k) < w2(i) (from monotonicity)
w2(k) < w2(j) L C R E j k i
graph G2

14. Proof outline Remove links from R to i and add links from L to i
w3(k) < w3(j) (from locality) k L C R E j i
graph G3

15. Proof outline Graphs G2 and G3 are the same up to a label permutationk L C R E j i k L C R E j i
graph G2
graph G3

16. Proof outline Graphs G2 and G3 are the same up to a label permutationk L C R E j i R C L E k j i
graph G2
graph G3

17. Proof outline We now have
w2(j) < w2(k) and w3(j) < w3(k) (shown before)
w2(j) = w3(k) and w2(k) = w3(j) (label independ.)
w2(j) > w2(k) CONTRADICTION! k L C R E j i R C L E k j i
graph G2
graph G3

18. Axiomatic characterization
All three properties are needed
locality
PageRank is also strictly monotone and label independent
monotonicity
consider an algorithm that assigns 1 to nodes with even degree, and 0 to nodes with odd degree
label independence
consider and algorithm that gives the more weight to links that come from some specific page (e.g. the Yahoo page)

19. Outline
Axiomatic Characterizations of Link Analysis Ranking Algorithms
InDegree algorithm
PageRank algorithm
Rank Aggregation
Computing aggregate scores
Computing aggregate rankings - voting

20. Self-edge axiom
Algorithm A satisfies the self-edge axiom if the following is true: If page a is ranked at least as high as page b in a graph G(V,E), where a does not have a link to itself, then a should be ranked higher than b in G(V,E U {v,v})

21. Vote by committee axiom
Algorithm A satisfies the vote by committee axiom if the following is true: If page a links to pages b and c, then the relative ranking of all the pages should be the same as in the case where the direct links from a to b and c are replaced by links from a to a new set of pages which link (only) to b and c

22. Vote by committee (example)

23. Collapsing axiom
If there is a pair of pages a and b that link to the same set of pages, but the set of pages that link to a and b are disjoint, then if a and b are collapsed into a single page (a), where links of b become links of a, then the relative rankings of all pages (except a and b) should remain the same.

24. Collapsing axiom (example)b c c

25. Proxy axiom
If there is a set of k pages with the same importance that link to a, and a itself links to k other pages, then by dropping a and connect the pages in N(a) and P(a), the relative ranking of all pages (excluding a) should remain the same

26. Proxy axiom (example) c

27. Axiomatic Characterization of PageRank Algorithm [AT04]
The PageRank algorithm satisfies label independence, self-edge, vote by committee, collapsing and proxy axioms.

28. Outline
Axiomatic Characterizations of Link Analysis Ranking Algorithms
InDegree algorithm
PageRank algorithm
Rank Aggregation
Computing aggregate scores
Computing aggregate rankings - voting

29. Rank Aggregation
Given a set of rankings R1,R2,…,Rm of a set of objects X1,X2,…,Xn produce a single ranking R that is in agreement with the existing rankings

30. Examples
Voting
rankings R1,R2,…,Rm are the voters, the objects X1,X2,…,Xn are the candidates.

31. Examples Combining multiple scoring functions
rankings R1,R2,…,Rm are the scoring functions, the objects X1,X2,…,Xn are data items.
Combine the PageRank scores with term-weighting scores
Combine scores for multimedia items
color, shape, texture
Combine scores for database tuples
find the best hotel according to price and location

32. Examples Combining multiple sources
rankings R1,R2,…,Rm are the sources, the objects X1,X2,…,Xn are data items.
meta-search engines for the Web
distributed databases
P2P sources

33. Variants of the problem
Combining scores
we know the scores assigned to objects by each ranking, and we want to compute a single score
Combining ordinal rankings
the scores are not known, only the ordering is known
the scores are known but we do not know how, or do not want to combine them
e.g. price and star rating

34. Combining scores Each object Xi has m scores (ri1,ri2,…,rim)
The score of object Xi is computed using an aggregate scoring function f(ri1,ri2,…,rim) R1 R2 R3 X1 1
0.3
0.2 X2
0.8 X3
0.5
0.7
0.6 X4 X5
0.1

35. Combining scores Each object Xi has m scores (ri1,ri2,…,rim)
The score of object Xi is computed using an aggregate scoring function f(ri1,ri2,…,rim)
f(ri1,ri2,…,rim) = min{ri1,ri2,…,rim} R1 R2 R3 R X1 1
0.3
0.2 X2
0.8 X3
0.5
0.7
0.6 X4 X5
0.1

36. Combining scores Each object Xi has m scores (ri1,ri2,…,rim)
The score of object Xi is computed using an aggregate scoring function f(ri1,ri2,…,rim)
f(ri1,ri2,…,rim) = max{ri1,ri2,…,rim} R1 R2 R3 R X1 1
0.3
0.2 X2
0.8 X3
0.5
0.7
0.6 X4 X5
0.1

37. Combining scores Each object Xi has m scores (ri1,ri2,…,rim)
The score of object Xi is computed using an aggregate scoring function f(ri1,ri2,…,rim)
f(ri1,ri2,…,rim) = ri1 + ri2 + …+ rim R1 R2 R3 R X1 1
0.3
0.2
1.5 X2
0.8
1.6 X3
0.5
0.7
0.6
1.8 X4
1.3 X5
0.1

38. Top-k
Given a set of n objects and m scoring lists sorted in decreasing order, find the top-k objects according to a scoring function f
top-k: a set T of k objects such that f(rj1,…,rjm) ≤ f(ri1,…,rim) for every object Xi in T and every object Xj not in T
Assumption: The function f is monotone
f(r1,…,rm) ≤ f(r1’,…,rm’) if ri ≤ ri’ for all i
Objective: Compute top-k with the minimum cost

39. Cost function
We want to minimize the number of accesses to the scoring lists
Sorted accesses: sequentially access the objects in the order in which they appear in a list
cost Cs
Random accesses: obtain the cost value for a specific object in a list
cost Cr
If s sorted accesses and r random accesses minimize s Cs + r Cr

40. Example Compute top-2 for the sum aggregate function R1 X1 1 X2 0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2

41. Fagin’s Algorithm
Access sequentially all lists in parallel until there are k objects that have been seen in all lists R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2

42. Fagin’s Algorithm
Access sequentially all lists in parallel until there are k objects that have been seen in all lists R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2

43. Fagin’s Algorithm
Access sequentially all lists in parallel until there are k objects that have been seen in all lists R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2

44. Fagin’s Algorithm
Access sequentially all lists in parallel until there are k objects that have been seen in all lists R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2

45. Fagin’s Algorithm
Access sequentially all lists in parallel until there are k objects that have been seen in all lists R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2

46. Fagin’s Algorithm
Perform random accesses to obtain the scores of all seen objects R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2

47. Fagin’s Algorithm Compute score for all objects and find the top-k R1X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2 R X3
1.8 X2
1.6 X1
1.5 X4
1.3

48. Fagin’s Algorithm
X5 cannot be in the top-2 because of the monotonicity property
f(X5) ≤ f(X1) ≤ f(X3) R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2 R X3
1.8 X2
1.6 X1
1.5 X4
1.3

49. Fagin’s Algorithm
The algorithm is cost optimal under some probabilistic assumptions for a restricted class of aggregate functions

50. Threshold algorithm Access the elements sequentially R1 X1 1 X2 0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2

51. Threshold algorithm At each sequential access
Set the threshold t to be the aggregate of the scores seen in this access R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2
t = 2.6

52. Threshold algorithm At each sequential access
Do random accesses and compute the score of the objects seen R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2
t = 2.6 X1
1.5 X2
1.6 X4
1.3

53. Threshold algorithm At each sequential access
Maintain a list of top-k objects seen so far R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2
t = 2.6 X2
1.6 X1
1.5

54. Threshold algorithm At each sequential access
When the scores of the top-k are greater or equal to the threshold, stop R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2
t = 2.1 X3
1.8 X2
1.6

55. Threshold algorithm At each sequential access
When the scores of the top-k are greater or equal to the threshold, stop R1 X1 1 X2
0.8 X3
0.5 X4
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2
t = 1.0 X3
1.8 X2
1.6

56. Threshold algorithm Return the top-k seen so far R1 X1 1 X2 0.8 X3 0.5
0.3 X5
0.1 R2 X2
0.8 X3
0.7 X1
0.3 X4
0.2 X5
0.1 R3 X4
0.8 X3
0.6 X1
0.2 X5
0.1 X2
t = 1.0 X3
1.8 X2
1.6

57. Threshold algorithm
From the monotonicity property for any object not seen, the score of the object is less than the threshold
f(X5) ≤ t ≤ f(X2)
The algorithm is instance cost-optimal
within a constant factor of the best algorithm on any database

58. Combining rankings In many cases the scores are not known
e.g. meta-search engines – scores are proprietary information
… or we do not know how they were obtained
one search engine returns score 10, the other 100. What does this mean?
… or the scores are incompatible
apples and oranges: does it make sense to combine price with distance?
In this cases we can only work with the rankings

59. The problem
Input: a set of rankings R1,R2,…,Rm of the objects X1,X2,…,Xn. Each ranking Ri is a total ordering of the objects
for every pair Xi,Xj either Xi is ranked above Xj or Xj is ranked above Xi
Output: A total ordering R that aggregates rankings R1,R2,…,Rm

60. Voting theory A voting system is a rank aggregation mechanism
Long history and literature
criteria and axioms for good voting systems

61. What is a good voting system?
The Condorcet criterion
if object A defeats every other object in a pairwise majority vote, then A should be ranked first
Extended Condorcet criterion
if the objects in a set X defeat in pairwise comparisons the objects in the set Y then the objects in X should be ranked above those in Y
Not all voting systems satisfy the Condorcet criterion!

62. Pairwise majority comparisons
Unfortunately the Condorcet winner does not always exist
irrational behavior of groups V1 V2 V3 1 A B C 2 3
A > B
B > C
C > A

63. Pairwise majority comparisons
Resolve cycles by imposing an agenda V1 V2 V3 1 A D E 2 B 3 C 4 5

64. Pairwise majority comparisons
Resolve cycles by imposing an agenda V1 V2 V3 1 A D E 2 B 3 C 4 5 A B A

65. Pairwise majority comparisons
Resolve cycles by imposing an agenda V1 V2 V3 1 A D E 2 B 3 C 4 5 A B A E E

66. Pairwise majority comparisons
Resolve cycles by imposing an agenda V1 V2 V3 1 A D E 2 B 3 C 4 5 A B A E E D D

67. Pairwise majority comparisons
Resolve cycles by imposing an agenda
C is the winner V1 V2 V3 1 A D E 2 B 3 C 4 5 A B A E E D D C C

68. Pairwise majority comparisons
Resolve cycles by imposing an agenda
But everybody prefers A or B over C V1 V2 V3 1 A D E 2 B 3 C 4 5 A B A E E D D C C

69. Pairwise majority comparisons
The voting system is not Pareto optimal
there exists another ordering that everybody prefers
Also, it is sensitive to the order of voting

70. Plurality vote Elect first whoever has more 1st position votes
Does not find a Condorcet winner (C in this case)
voters 10 8 7 1 A C B 2 3

71. Plurality with runoff
If no-one gets more than 50% of the 1st position votes, take the majority winner of the first two
voters 10 8 7 2 1 A C B 3
first round: A 10, B 9, C 8
second round: A 18, B 9
winner: A

72. Plurality with runoff
If no-one gets more than 50% of the 1st position votes, take the majority winner of the first two
voters 10 8 7 2 1 A C B 3
change the order of
A and B in the last
column
first round: A 12, B 7, C 8
second round: A 12, C 15
winner: C!

73. Positive Association axiom
Plurality with runoff violates the positive association axiom
Positive association axiom: positive changes in preferences for an object should not cause the ranking of the object to decrease

74. Borda Count
For each ranking, assign to object X, number of points equal to the number of objects it defeats
first position gets n-1 points, second n-2, …, last 0 points
The total weight of X is the number of points it accumulates from all rankings

75. Borda Count Does not always produce Condorcet winner 3 2 1 (3p) A B C
voters 3 2
1 (3p) A B C
2 (2p) D
3 (1p)
4 (0p) BC C B A D
A: 3*3 + 2*0 + 2*1 = 11p
B: 3*2 + 2*3 + 2*0 = 12p
C: 3*1 + 2*2 + 2*3 = 13p
D: 3*0 + 2*1 + 2*2 = 6p

76. Borda Count Assume that D is removed from the vote
Changing the position of D changes the order of the other elements!
voters 3 2
1 (2p) A B C
2 (1p)
3 (0p) BC B A C
A: 3*2 + 2*0 + 2*1 = 7p
B: 3*1 + 2*2 + 2*0 = 7p
C: 3*0 + 2*1 + 2*2 = 6p

77. Independence of Irrelevant Alternatives
The relative ranking of X and Y should not depend on a third object Z
heavily debated axiom

78. Borda Count
The Borda Count of an an object X is the aggregate number of pairwise comparisons that the object X wins
follows from the fact that in one ranking X wins all the pairwise comparisons with objects that are under X in the ranking

79. Voting Theory
Is there a voting system that does not suffer from the previous shortcomings?

80. Arrow’s Impossibility Theorem
There is no voting system that satisfies the following axioms
Universality
all inputs are possible
Completeness and Transitivity
for each input we produce an answer and it is meaningful
Positive Assosiation
Independence of Irrelevant Alternatives
Non-imposition
Non-dictatoriship
KENNETH J. ARROW Social Choice and Individual Values (1951). Won Nobel Prize in 1972

81. Kemeny Optimal Aggregation
Kemeny distance K(R1,R2): The number of pairs of nodes that are ranked in a different order (Kendall-tau)
number of bubble-sort swaps required to transform one ranking into another
Kemeny optimal aggregation minimizes
Kemeny optimal aggregation satisfies the Condorcet criterion and the extended Condorcet criterion
maximum likelihood interpretation: produces the ranking that is most likely to have generated the observed rankings
…but it is NP-hard to compute
easy 2-approximation by obtaining the best of the input rankings, but it is not “interesting”

82. Locally Kemeny optimal aggregation
A ranking R is locally Kemeny optimal if there is no bubble-sort swap that produces a ranking R’ such that K(R’,R1,…,Rm)≤ K(R’,R1,…,Rm)
Locally Kemeny optimal is not necessarily Kemeny optimal
Definitions apply for the case of partial lists also

83. Locally Kemeny optimal aggregation
Locally Kemeny optimal aggregation can be computed in polynomial time
At the i-th iteration insert the i-th element x in the bottom of the list, and bubble it up until there is an element y such that the majority places y over x
Locally Kemeny optimal aggregation satisfies the Condorcet and extended Condorcet criterion

84. Rank Aggregation algorithm [DKNS01]
Start with an aggregated ranking and make it into a locally Kemeny optimal aggregation
How do we select the initial aggregation?
Use another aggregation method
Create a Markov Chain where you move from an object X, to another object Y that is ranked higher by the majority

85. Spearman’s footrule distance
Spearman’s footrule distance: The difference between the ranks R(i) and R’(i) assigned to object i
Relation between Spearman’s footrule and Kemeny distance

86. Spearman’s footrule aggregation
Find the ranking R, that minimizes
The optimal Spearman’s footrule aggregation can be computed in polynomial time
It also gives a 2-approximation to the Kemeny optimal aggregation
If the median ranks of the objects are unique then this ordering is optimal

87. Example R1 1 A 2 B 3 C 4 D R2 1 B 2 A 3 D 4 C R3 1 B 2 C 3 A 4 D R 1 B

88. The MedRank algorithm Access the rankings sequentially R1 1 A 2 B 3 C4 D R2 1 B 2 A 3 D 4 C R3 1 B 2 C 3 A 4 D R 1 2 3 4

89. The MedRank algorithm Access the rankings sequentially
when an element has appeared in more than half of the rankings, output it in the aggregated ranking R1 1 A 2 B 3 C 4 D R2 1 B 2 A 3 D 4 C R3 1 B 2 C 3 A 4 D R 1 B 2 3 4

90. The MedRank algorithm Access the rankings sequentially
when an element has appeared in more than half of the rankings, output it in the aggregated ranking R1 1 A 2 B 3 C 4 D R2 1 B 2 A 3 D 4 C R3 1 B 2 C 3 A 4 D R 1 B 2 A 3 4

91. The MedRank algorithm Access the rankings sequentially
when an element has appeared in more than half of the rankings, output it in the aggregated ranking R1 1 A 2 B 3 C 4 D R2 1 B 2 A 3 D 4 C R3 1 B 2 C 3 A 4 D R 1 B 2 A 3 C 4

92. The MedRank algorithm Access the rankings sequentially
when an element has appeared in more than half of the rankings, output it in the aggregated ranking R1 1 A 2 B 3 C 4 D R2 1 B 2 A 3 D 4 C R3 1 B 2 C 3 A 4 D R 1 B 2 A 3 C 4 D

93. The Spearman’s rank correlation
Computing the optimal rank aggregation with respect to Spearman’s rank correlation is the same as computing Borda Count
Computable in polynomial time

94. Extensions and Applications
Rank distance measures between partial orderings and top-k lists
Similarity search
Ranked Join Indices
Analysis of Link Analysis Ranking algorithms
Connections with machine learning

95. References
A. Borodin, G. Roberts, J. Rosenthal, P. Tsaparas, Link Analysis Ranking: Algorithms, Theory and Experiments, ACM Transactions on Internet Technologies (TOIT), 5(1), 2005
Ron Fagin, Ravi Kumar, Mohammad Mahdian, D. Sivakumar, Erik Vee, Comparing and aggregating rankings with ties , PODS 2004
M. Tennenholtz, and Alon Altman, "On the Axiomatic Foundations of Ranking Systems", Proceedings of IJCAI, 2005
Ron Fagin, Amnon Lotem, Moni Naor. Optimal aggregation algorithms for middleware, J. Computer and System Sciences 66 (2003), pp Extended abstract appeared in Proc ACM Symposium on Principles of Database Systems (PODS '01), pp
Alex Tabbarok Lecture Notes
Ron Fagin, Ravi Kumar, D. Sivakumar Efficient similarity search and classification via rank aggregation, Proc ACM SIGMOD Conference (SIGMOD '03), pp
Cynthia Dwork, Ravi Kumar, Moni Naor, D. Sivakumar. Rank Aggregation Methods for the Web. 10th International World Wide Web Conference, May 2001.
C. Dwork, R. Kumar, M. Naor, D. Sivakumar, "Rank Aggregation Revisited," WWW10; selected as Web Search Area highlight, 2001.